Almost Markovian processes from closed dynamics

Almost Markovian processes from closed dynamics

Pedro Figueroa–Romero, Kavan Modi, and Felix A. Pollock

School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia

April 30, 2019

It is common, when dealing with quan-tum processes involving a subsystem of a much larger composite closed system, to treat them as effectively memory-less (Markovian). While open systems theory tells us that non-Markovian processes should be the norm, the ubiquity of Markovian processes is undeniable. Here, without resorting to the Born-Markov assumption of weak coupling or making any approximations, we formally prove that pro-cesses are close to Markovian ones, when the subsystem is sufficiently small compared to the remainder of the composite, with a probabil-ity that tends to unprobabil-ity exponentially in the size of the latter. We also show that, for a fixed global system size, it may not be possi-ble to neglect non-Markovian effects when the process is allowed to continue for long enough. However, detecting non-Markovianity for such processes would usually require non-trivial en-tangling resources. Our results have founda-tional importance, as they give birth toalmost

Markovian processes from composite closed dynamics, and to obtain them we introduce a new notion of equilibration that is far stronger than the conventional one and show that this stronger equilibration is attained.

1

Introduction

The quest towards understanding how thermodynam-ics emerges from quantum theory has seen a great deal of recent progress [1,2]. Most prominently, the conun-drum of how to recover irreversible phenomena that obey the second law of thermodynamics, starting from reversible and recurrent Schrödinger dynamics, has been rectified by considering an analogous dynamical equilibrium. This is known generically asequilibration on average; it implies that time-dependent quantum properties evolve towards a certain fixed equilibrium value, and stay close to it for most times1.

Equilibration on average has been widely studied

Pedro Figueroa–Romero: [email protected]

1_{Even stronger (but usually the one of interest) is the notion}

of equilibration in a finite time interval. The corresponding statement for a physical system replaces the term equilibration withthermalisationand the equilibrium state with athermal state. For detail see e.g. [1]

−→

e−iHτ

t= 0 t=τ E

S

E S

(a)

hO

iρ(

t

)

−

hO

iω

t 0

(b)

· · · ρ

E

S

M0 U1

M1

U2 · · ·

Mk

−1 Uk

\

ρ(_kS)

(c)

Figure 1: Weak and strong equilibration. (a) A notion of subsystem equilibration: S is in a nonequilibrium state at time0and evolves jointly withEaccording to Schrödinger’s equation until it reaches an equilibrium state at some time

τ; more precisely, in (b) the expectationh · iof an observable

Oon the stateρ(t)differs only slightly from the correspond-ing one on the time-averaged state ωfor most times, even if it eventually deviates greatly from it. In (c) we illustrate the generalization we consider in this work, where an initial joint stateρevolves unitarily between the application of mea-surementsMi on the systemS, rendering the final reduced stateρ(_kS); the dotted box encloses the process tensor which describes the process independently of the (experimentally) controllable operations.

for small non-degenerate subsystems of larger closed systems, which are usually taken to be (quasi) iso-lated, and for particular models (see e.g. [3]). In a typical setup, the closed dynamics of a system and environment, as illustrated in Fig.1(a), is considered. Remarkably, it can be shown that the expectation value for any observableO on the system alone lies in the neighbourhood of its time-averaged value for almost all times [Fig. 1(b)]. Yet, such equilibration results, in full generality, remain unsolved [1]; in par-ticular, it is unknown whether they extend to corre-lations between observables at different times.

To see this, consider the case depicted in Fig.1(c), where the subsystem is interrogated (by measuring, or otherwise manipulating it) multiple times as it evolves with the remainder of the closed system, which acts as an environment. In such a scenario, the flow of information between interventions must be taken into

ρ(1) E

ρ_S

E

S

M0

U1

\

ρ(2) E

M1

U2

\

ρ(k) E

Mk

−1 Uk

\

ρ(_kS)

Figure 2: Markovian dynamics. A quantum circuit corre-sponding to the Markov approximation, which is mathemati-cally equivalent to starting off with an uncorrelatedρ(1)

E ⊗ρS

state and then artificially refreshing the environment (dis-carding and replacing with a new state) at each step (cf. Fig. 1c)

account; in particular, there may be intrinsic tempo-ral correlations mediated by the environment, in other wordsmemory built up within the dynamics itself, in-dependently of the external interventions, which are the hallmark of so-called non-Markovian dynamics. Equilibration on average says that the process erases the information contained in the initial state of the system; however, there may still be non-Markovian memory of the initial state encoded in the tempo-ral correlations between observables. It is therefore an open question whether this information is truly scrambled by the process and therefore inaccessible to the system at long times. Such memoryless dynam-ics are anticipated for the subparts of closed many-body dynamics [4]. This scrambling of information in correlations would represent a stronger notion of equilibration than the usual one, which only consid-ers statistics of individual observables at a given time. Moreover, the absence of memory effects would gener-ically imply the usual kind of equilibration.

Searching for this stronger notion of equilibration is not physically unmotivated. While, strictly speaking, non-Markovian dynamics is the norm for subsystems of a closed system evolving with a time-independent Hamiltonian [5, 6], Markov approximations are com-monly made throughout physics (and elsewhere) to good effect. However, the standard approach to mak-ing this approximation in open quantum systems the-ory involves a series of assumptions about weak intactions with an environment sufficiently large and er-godic that the memory of past interactions gets prac-tically lost [7, 8]; this is mathematically equivalent (through an analogue of Stinespring’s dilation the-orem) to continually refreshing (discarding and re-placing) the environment’s state [9], i.e., artificially throwing away information from the environment (see Fig. 2). In reality, this is not usually how a closed system evolves, leading to the question: In the gen-eral case, how does dissipative Markovian dynamics emerge from closed Schrödinger evolution? That this puzzle is similar to the ones concerning equilibra-tion and thermalisaequilibra-tion should not be too surpris-ing, as thermalisation is often achieved by assuming a Markov process [10].

Here, we give an initial answer to this deep founda-tional question. We show that, for the class of

evolu-tions given by the Haar measure (typically involving an interaction of the subsystem with a significant por-tion of the remainder), above a critical time scale the dynamics of a subsystem is exponentially close to a Markov process as a function of the relative size of the subsystem compared with the whole. Moreover, our result does not assume weak coupling between the system and the environment, and we do not em-ploy any approximations. Specifically, we use aver-ages over the unitary group together with the con-centration of measure phenomenon [11–15] to deter-mine when a subsystem dynamics might typically be close to Markovian. Our approach is broadly similar to that of Refs. [14, 16] both in spirit and in terms of the mathematical tools employed. In other words, our main result gives a mechanism for equilibration starting from closed dynamics without employing any approximations.

Our results are possible thanks to recent efforts towards the understanding of general quantum pro-cesses, culminating in the development of the process tensor framework [17–19], which allows for the com-plete and compact description of an open system’s evolution under the influence of a series of external manipulations or measurements [20,21].

Crucially, the process tensor framework leads to an unambiguous distinction between Markovian and non-Markovian dynamics, analogous to the usual one for classical stochastic processes. In this sense, it subsumes other approaches to quantum non-Markovianity (when compared with respect to the same process), including those based on increasing trace-distance distinguishability [22]. It allows for a precise quantification of memory effects (including across multiple time points) [5], that we employ here as a natural setting to explore temporal correlations and the role that they play in a generalized, or strong, notion of equilibration.

2

Quantum processes with memory:

the process tensor

We begin with the motivation of witnessing equili-bration effects through a sequence of measurements on the system. We consider a closed environment-system (ES) with finite dimensional Hilbert space

HES =CdE_⊗

CdS _{initialised in a pure joint stateρ.} The systemSis acted on with an initial measurement operationM0followed by a series of subsequent

any other experimental manipulations of the system, but we focus on the former in order to be concrete. This scenario is depicted in Fig. 1(c). Formally, ak -step process tensor is a completely positive causal 2 map Tk:0 from the set {Mi}ki=0−1 to output states,

ρ0_k=Tk:0[{Mi}ki=0−1].

It must be stressed, however, that although experi-mental interventions play an important role in the re-construction of a process through a generalized quan-tum process tomography, they do not determine its nature, i.e., the process tensor itself accounts precisely for the intrinsic dynamics occurring independently of the control of the experimenter. Moreover, like any other CP map, the process tensor can be expressed as a (many-body) quantum state through the Choi-Jamiołkowski isomorphism [21, 23, 24]. For a k-step process tensor, the corresponding (normalized) Choi state Υk:0 can be obtained by swapping the system

with one half of a maximally entangled stateΨAiBi ≡ |ψ+_ihψ+_{| ≡}PdS

j,`=1|jjih``|/dSat each timei[25]. The

2k ancillary systems A1, B1,· · ·Ak, Bk have Hilbert

space dimension dim(HA) = dim(HB) = dS 3 and

the system is swapped with an Asubsystem at each time. With this convention, the Choi state of the pro-cess tensor can be seen to take the form

Υk:0= trE[Uk:0(ρ⊗Ψ⊗k)Uk†:0],

with Uk:0≡(Uk⊗1)Sk· · ·(U1⊗1)S1(U0⊗1),

(1)

where all identity operators act on the ancillary sys-tems,Ui is theES unitary operator that evolves the

joint system following the control operation at timei and theSi are swap operators between systemSand the ancillary systemAi. The corresponding quantum

circuit is depicted in Fig.3.

ρ E

S U0 U1 · · · Uk

\

· · ·

ΨA1B1

· · ·

ΨA2B2

.. .

· · ·

.. .

ΨAkBk

Υk

:0 ×

S1 ×

×

S2 ×

×

Sk

×

A1

B1

A2

B2

Ak

Bk

Figure 3: Choi state of a process tensor. A quantum circuit diagram for the Choi state of ak-step process tensor under jointES unitary evolution: the final stateΥis a many-body quantum state capturing all the properties of the process.

2_{Meaning that the output of a process tensor at timekdoes}

not depend on deterministic inputs at later timesK > k.

3 _{The dimensions of eachA}

i,Bi must be fixed to dS to make the swap operator well defined.

Throughout the paper, we write Υ for the Choi state of ak-step process unless otherwise specified.

3

An unambiguous measure of

non-Markovianity

Among other important properties, the process ten-sor leads to a well-defined Markov criterion, from which it is possible to construct a family of oper-ationally meaningful measures of non-Markovianity, many of which can be stated simply as distances

between a process tensor’s Choi state Υ and that of a corresponding Markovian one Υ(M)_{. The latter}

must take the form of a tensor product of quantum maps Ei:i−1 connecting adjacent pairs of time steps:

Υ(M)₌Nk

i=1Ei:i−1 [25].

In [5], a such measure of non-Markovianity was first introduced through the relative entropy be-tween a process and its corresponding Markovian one,

R(ΥkΥ(M)_{)≡tr[Υ(log Υ−log Υ}(M)_{)], minimized over}

the latter. Here, however, in analogy with other stud-ies on equilibration, we choose the measure of non-Markovianity defined in terms of the trace distanceD

as

N ≡min

Υ(M)

D(Υ,Υ(M)_)≡1

2Υmin(M)

kΥ−Υ(M)_k

1 (2)

where kXk1≡tr

√

XX† _{is the trace norm (or}

Schat-ten 1-norm); in particular, this is related to relative entropy through the so-called quantum Pinsker in-equality, R(ΥkΥ(M)_)≥2D2_(Υ,Υ(M)_{). The trace}

dis-tance is a natural choice, as it represents an impor-tant distinguishability measure and has played a cen-tral role in the discovery of several major results on typicality [14] and equilibration [1, 16, 26] (or lack thereof, e.g. [27]). It has an experimental importance for comparing quantum processes [28] and is naturally related to other important distinguishability measures such as the diamond norm or the fidelity. However, it is important to point out that the above measure is fundamentally different (and more powerful) than other measures of non-Markovianity that have been proposed in recent years, including one that employs trace distance [22]. The measure we use here cap-turesall non-Markovian features across multiple time steps and provides an operational meaning for non-Markovianity. Equipped with this measure, we are in a position to determine what value it takes for a typical process, answering the question posed at the beginning of this manuscript.

4

Typicality of Markovian processes

which one could sample from the space of quantum processes, but as we are particularly interested in the typicality (or absence thereof) of Markovian pro-cesses independently of ad-hoc assumptions like weak coupling, we would like our measure to assign non-vanishing probabilities to mathematically generic uni-tary dynamics on the closedES system.

Here, we achieve this by sampling the evolution from the unitarily invariant probability measure, the so-called Haar measure. This has the additional ad-vantage of allowing us to average by means of ran-dom matrix theory techniques [15, 29–32] and leads to the relatively straightforward application of con-centration of measure results [11–13]. Specifically, we use the Haar measure to sample two distinct types of ES evolution:

1. Random interaction: AllUiindependently chosen,

2. Constant interaction: Ui=Uj, ∀ ≤i6=j≤k,

where Ui represents the closed (unitary) ES

evolu-tion between the applicaevolu-tion of measurement opera-tionMi−1andMi, as depicted in Fig.1(c); the entire

set enters into the process tensor through Eq.(1). In the first case, the global system will quickly explore its entire (pure) state space for any initial state. The sec-ond case correspsec-onds more closely to what one might expect for a truly closed system, where the Hamilto-nian remains the same throughout the process. These correspond to two extremes; more generally, the dy-namics from step to step may be related but not iden-tical.

We highlight that under this sampling method, and thus this notion of typicality, the interaction or infor-mation flow between all parts of the whole system plus environment is relevant, i.e. no parts of the environ-ment dimension are superfluous. We will now show that this kind of dynamics is almost always close to Markovian.

4.1

Main result

Our main result states that, for a randomly sam-pled k-step quantum process Υ undergone by a dS

-dimensional subsystem of a larger dEdS-dimensional

composite, the probability for the non-Markovianity

N to exceed a function ofk,dS anddE, that becomes

very small in the largedE limit, itself becomes small

in that limit. Precisely, we prove that, for an arbitrary >0,

P[N ≥ Bk(dE, dS) +]≤e−C(dE,dS) 2

, (3)

whereC(dE, dS) =c dEdS

_d

S−1 dk_S+1−1

2

withc= 1/4for

a constant interaction process andc= (k+ 1)/4for a random interaction process. The functionBk(dE, dS)

is an upper bound on the expected non-Markovianity

E[N]whose details depend on the way in which pro-cesses are sampled.

Υ(M)

Υ

Bk E[N] N

(a)

Υ(M)

Υ

(b)

Figure 4: Concentration of Markovian processes in large environments. (a) A geometric cartoon of our main result in a space of process tensors: the probability of the non-Markovianity N of deviating from Bk by some > 0 de-creases exponentially in 2. In (b), quantum processes Υ

on large dimensional environments (such thatdEd2Sk+1) concentrate around the Markovian onesΥ(M)

.

The proof is presented in full in AppendixE, and it can be outlined as follows: we first bound the differ-ence between N for two different processes in terms of the distance between the unitaries used to gener-ate them; Levy’s lemma [11,13, 14] then states that the fraction of processes with non-Markovianity more than away from the expectation value (which we upper-bound by Bk) is bounded by a concentration function. By considering the geometry of the spaces of unitaries we are sampling from, we are able to use the exponential function appearing on the right hand side of Eq.(3). This implies that our result holds par-ticularly for this class of evolutions where all parts of system and environment effectively interact, in con-trast with some commonly considered open systems models.

Our result is meaningful when bothBk+ande−C2 are small; the latter is fulfilled in thesmall subsystem

orlarge environment limit, which in our setting means dEd2Sk+1. We may also state the minimal value for

that, assuming the large environment limit, renders both sides small, i.e., such that 2_d

E 1 ; this

is fulfilled for =d−1_E /3 4_{. A geometrical cartoon to} illustrate the result is presented in Fig. 4.

This is the case because the upper bound Bk ≥

4_{Similar to [14], here we look for anx >0 such that=d}−x E and2_d

E(N)is given by

Bk(dE, dS)≡



 

 

√

dEE[tr(Υ2_)]−x+y

2 if dE< d

2k+1

S

p

d2_Sk+1E[tr(Υ2_)]−1

2 if dE≥d

2k+1

S ,

(4)

withx≡ dE

d2_Sk+1(1 +y),y≡1− dE

d2_Sk+1. The size ofBk

depends entirely on the size of the average purity of the process tensor_E[tr(Υ2_{)], which we have computed}

analytically for the case of Haar-randomly sampled evolutions (which equivalently could be sampled from (2k+ 2)-unitary designs [33, 34], as we will discuss below) in AppendixF.

The purity tr(Υ2) is a quantifier of the mixed-ness (uniformity of eigenvalues) of a positive opera-tor. When computed on reduced states of bipartite systems it can also serve as a quantifier of entangle-ment. In Appendix D, we analytically compute the expected purity of the Choi state of a process tensor

E[Υ2] in both the constant and the random interac-tion pictures, which can be directly translated as a quantifier for noisiness of the quantum process itself and entanglement between system and environment. The expressions take the form

E_Ui[tr(Υ2)] = d

2

E−1

dE(dEdS+ 1)

_d2

E−1

d2

Ed

2

S−1

k

+ 1 dE

,

(5)

for the random interaction picture and

EU[tr(Υ

2_{)] =d}−2k

S

X

σ,τ∈S2k+2

Wg(τ σ−1)%τ∆

(dE,dS) k,σ,τ , (6)

in the constant interaction case, whereSn is the sym-metric group over n elements, Wg is known as the Weingarten function [30],%τ is a product of entries of

ρ depending on permutationsτ and ∆ is a product, scaling withk, of monomials indE anddS depending

on permutationsσandτ.

4.2

Limiting cases

In both the constant and random interaction cases,

Bk is a well-behaved rational function ofdE, dS and

k. Eq. (6) takes a non-trivial form mainly because of the Wg function (which is intrinsic to the Haar-unitary averaging in the constant interaction picture). However, due to results found in [30,31], we can still study analytically the behaviour of the boundBk for both cases in the two following limits.

In the large environment limit,

lim

dE→∞

E[tr(Υ2)] = 1

d2_Sk+1, (7)

which corresponds to the purity of the maximally mixed state. Note that the averaging occurs after

computing the purity; that is, independently of which case is considered. Every process sampled will be in-distinguishable from the maximally noisy (and hence Markovian) one, with probability that tends to one as dE is increased:

lim

dE→∞

E[N] = lim

dE→∞

Bk(dE, dS) = 0. (8)

Furthermore, from Eq.(5)it is seen that it does so at a rate O(1/dE)in the random interaction case.

The other interesting limiting case is the one where the ES dimension is fixed, but the number of time steps is taken to be very large. The resulting process encodes all high order correlation functions between observables over a long period of time. Again for both cases, the expected purity in this limit goes as

lim

k→∞E[tr(Υ

2_{)] =} 1

dE

, (9)

which corresponds to the maximally mixed purity of the environment. This implies that the Choi state of the full ES unitary process is maximally entangled between S and E. In this limit, we get correspond-ingly

lim

k→∞E[N]≤klim→∞Bk(dE, dS) = 1, (10)

meaning that nothing can be said about typical non-Markovianity; a typical process in this limit could be highly non-Markovian. Indeed, we expect this to be the case, since the finite-dimensional ES space will have a finite recurrence time.

Both our main result in Eq.(3), and the bound on non-Markovianity, Eq. (4), generalize well-known re-sults for quantum states, i.e., when k = 0. In this case, our main result reduces to the usual one for the typicality of maximally mixed states (or bipar-tite maximally entangled states), with C(dE, dS) =

dEdS/4 (see e.g. [35]). As we also detail in

Ap-pendix F, when we take the Haar measure, in either case, we recover the average purity

E[tr(ρ2S)] =

dE+dS

dEdS+ 1

, (11)

where ρS ≡Υ0:0 = trE(U ρ U†), as found in [15,36– 41]. This then leads to

E

D

ρS, 1

dS

≤ 1

2

s

d2

S−1

dEdS+ 1

, (12)

which goes aspdS/4dE whendEdS, and is also a

standard result [35].

4.3

Numerical examples

●

●

● ●

● ●

● ● ● ●

● ● ● ● ● ●

■

■

■ ■

■ ■

■ ■

■

■ ■ ■ ■ ■ ■ ■

◆ ◆

◆ ◆

◆ ◆

◆ ◆

◆ ◆

◆ ◆

◆ ◆

◆ ◆

● k=1 ■ k=2

◆ k=3

22₂3 ₂4 ₂5 ₂6 ₂7

0.0 0.2 0.4 0.6 0.8 1.0

dE

U

i

[



]

Figure 5: Average non-MarkovianityEUi[N]of a random

interaction process for a qubit in the environment dimen-siondE at fixed time stepsk. Discrete values are shown for numerical averages overb40/kcrandomly generated process tensorsΥat time stepsk= 1,2,3and with fixeddS= 2; er-ror bars denote the standard deviation due to sampling erer-ror. The lines above each set of points denote the upper bound

Bk(dE,2). Process tensors were generated by sampling Haar random unitaries according to [42].

E[N]as a function of environment dimension dE for

a fixed system dimension dS = 2, obtaining the

be-haviour shown in Fig. 5. For constant interaction the numerical results are practically indistinguishable from those in the random case, but as mentioned, the analytical bound Bk is much harder to compute actly. This suggests that either a simpler bound ex-ists or that it might be possible to simplify the one we have obtained. As expected, our numerical re-sults fall within the bound Bk(dE, dS = 2) and they

behave similarly; we notice that the bound in gen-eral seems to be somewhat loose, and become loosest when dE ' d2Sk+1, implying that non-Markovianity

might be hard to detect even when not strictly in the large environment limit. However, it does saturate rapidly ask increases.

So far, our results are valid for process tensors constructed with Haar random unitaries at k evenly spaced steps; we are effectively considering a strong interaction between system and environment which rapidly scrambles quantum information in both [33]. The mechanism for equilibration is precisely that of dephasing [1], or effectively, the scrambling of infor-mation on the initial state of the system. This sug-gests that, even when timescales will differ with the type of evolution considered, most physical evolutions fall within our result, with e.g. a weaker behavior in k.

Our results could potentially be extended to their analogues sampled from unitary n-designs, approxi-mate or otherwise [43], that reproduce the Haar dis-tribution up to the n-th moment. As mentioned above, our expression for the upper bound in Eq.(4) is identical for samples from a2k+ 2-designs, and, if the non-Markovianity N could be approximated by

a polynomial function of U, Eq. (3) would also hold for n-designs with large enough n[44]. This type of distribution has recently been shown to be approx-imated by a wide class of physical evolutions [45]. In particular, the unitary circuits involved in com-plex quantum computations typically look Haar ran-dom [34]; if it holds in this case, our result would therefore suggest that the behaviour of components of a quantum computer would not typically exhibit non-Markovian memory (at least due to the compu-tation itself). Furthermore, even when it’s known that ensembles {e−iHt_}t

≥0 for time-independent

Hamilto-nians H cannot strictly become Haar random [33], certain complex Hamiltonian systems can be treated as random up to a certain Haar-moment via unitary designs [45].

These issues should be investigated further, how-ever, we will now show that our results still hold at a coarse-grained level, where the intermediate dynam-ics corresponds to products of Haar random unitaries, which are not themselves Haar random.

5

Observing non-Markovianity

The choice of unitaries going into the process ten-sor in the previous sections (in our case drawn from the Haar measure), dictates a time scale for the sys-tem, up to a freely chosen energy scale. However, we can straightforwardly construct process tensors on a longer, coarse-grained time scale by simply allowing the system to evolve (with an identity operation) be-tween some subset of time steps. In fact, process ten-sors at all time scales should be related in this way to an underlying process tensor with an infinite number of steps [46].

To see that our main result directly applies to any process tensor that can be obtained through coarse graining, we again consider the definition of our non-Markovianity measure, given in Eq.(2). Consider the coarser grained process tensor Υk:0/{i∈[0,k−1]} ≡Υc,

where a subset of control operations {Mi}i∈[0,k−1]

are replaced by identity operations (i.e., the sys-tem is simply left to evolve). Letting Ncoarse ≡ min_Υ(M)

c D(Υc,Υ

(M)

c ), we have

Ncoarse≤ N, (13)

since the set of allowed Υ(M)

c strictly contains the

al-lowed Υ(M) _{at the finer-grained level. This renders}

the new process less distinguishable from a Marko-vian one, i.e., coarse-graining can only make processes more Markovian.

due to the same reasoning, we cannot say anything about finer-grained dynamics. One approach to tack-ling this issue would be to choose a different samptack-ling procedure which explicitly takes scales into account. However, we leave this for future work.

There is another important limitation for observing non-Markovianity. The operational interpretation of the trace distance, discussed in the previous section, implies that observing non-Markovianity requires ap-plying a measurement that is an eigen-projector op-erator ofΥ−Υ(M)_{. The optimal measurement will, in}

general, be entangled across all time steps. In prac-tice, this is hard to achieve and typically one considers a sequence of local measurements m ∈ M. In gen-eral, for an any set of measurementsMwe can define a restricted measure of non-Markovianity detectable with that set: D_M(Υ,Υ(M)_{) ≡ max}

m∈M

1

2|tr[m(Υ−

Υ(M)_{)]| ≤ D(Υ,Υ}(M)_{), which means that the}

de-tectable non-Markovianity will be smaller. This is akin to theeigenstate thermalization hypothesis [47], where all eigenstates of a physical Hamiltonian look uniformally distributed with respect to most ‘phys-ically reasonable’ observables. In our setting, this means that looking for non-Markovianity with observ-ables that are local in time – i.e., ‘physically reason-able’ – we find almost no temporal correlations.

The locality constraint, along with monotonicity of non-Markovianity under coarse graining, have fur-ther important consequences for a broad class of open systems studies where master equations are em-ployed [48]. Since master equations usually only ac-count for two-point correlations with local measure-ments, they will be insensitive to most of the tempo-ral correlations being accounted for by our measure, leading to an even greater likelihood for their descrip-tions to be Markovian. We now discuss the broader implications of our results.

6

Discussions and Conclusions

Our results imply that it is fundamentally hard to ob-serve non-Markovianity in a typical process and thus go some way to explaining the overwhelming success of Markovian theories.

Specifically, we have shown that, even when inter-acting strongly with the wider composite system, a subsystem will typically undergo highly Markovian dynamics when the rest of the system has a suffi-ciently large dimension, and that the probability to be significantly non-Markovian vanishes with the lat-ter. Our main result formalizes the notion that in the large environment limit a quantum process, taken uni-formly at random, will be almost Markovian with very high probability. This corroborates the common un-derstanding of the Born-Markov approximation [7,8], but, crucially, we make no assumptions about weak coupling betweenE andS. Instead, in the Haar ran-dom interactions we consider, every part of the

sys-tem typically interacts significantly with every part ofE. This is in contrast to many open systems mod-els, even those with superficially infinite dimensional baths, where the effective dimension of the environ-ment is relatively small [49]; it can always be bounded by a function of time scales in the system-environment Hamiltonian [50], which could be encoded in a bath spectral density. Our result is also more general than the scenario usually considered, since it accounts for interventions and thus the flow of information be-tween S andE across multiple times.

While it may still be possible to observe non-Markovian behaviour at a time scale that is smaller than the fundamental time scale set by the chosen uni-taries, Eq. (13)tells us that any coarse grained pro-cess will remain concentrated around the Markovian ones in the large environment limit. Otherwise, for larger and larger systems, one needs an ever increas-ing number of time steps, correspondincreas-ing to higher or-der correlations, in oror-der to increase the probability of witnessing non-Markovianity. However, even in this case, from the discussion in the previous section, we know that the measurement on this large number of times steps will be temporally entangled, which may also be difficult to achieve.

We have introduced a stronger notion of equilibra-tion than the convenequilibra-tional one [1], so that strong equi-libration implies weak equiequi-libration. Our main result shows that it is possible to attain strong equilibration, where the information of the initial state of the sys-tem does not exist in multi-time correlations, much less expectation values at different times. Our result can be interpreted as a mechanism for equilibration and thermalisation. That is, we have shown that the dynamics of the system alone is almost Markovian, and such processes have well defined fixed points [51]. Even if we do not characterise the equilibrium state here, e.g. as done for quantum states under random-ized local Hamiltonians [52], we may conclude that the state of the system approaches this fixed point and the expectation value for any observable will lie in the neighbourhood of the time-averaged value [as depicted in Fig. 1(b)] for most times. However, one point of departure between the two results is our re-liance on Haar sampling.

More precisely, we have considered the case where the joint unitary evolution between each time step is drawn uniformly at random according to the Haar measure, for the two limiting cases of the random uni-tary operators at each step being the same or inde-pendent. This is, however, different from simply sam-pling the Choi state of the process Haar randomly. It is also not equivalent to picking Haar random states for the system at every time, as one would expect if the results of Ref. [14] were to hold independently at all times.

potentially be made precise and other issues such as equilibration time scales can also be readily explored; the approach that we have presented constitutes a first significant step towards achieving this goal.

Acknowledgments

We are grateful to Simon Milz and Andrea Collevec-chio for valuable discussions. PFR is supported by the Monash Graduate Scholarship (MGS) and the Monash International Postgraduate Research Schol-arship (MIPRS). KM is supported through Australian Research Council Future Fellowship FT160100073.

References

[1] C. Gogolin and J. Eisert, “Equilibration, ther-malisation, and the emergence of statistical me-chanics in closed quantum systems,” Rep. Prog. Phys.79, 056001 (2016).

[2] J. Goold, M. Huber, A. Riera, L. del Rio, and P. Skrzypczyk, “The role of quantum information in thermodynamics–a topical review,”J. Phys. A: Math. Theor.49, 143001 (2016).

[3] L. P. García-Pintos, N. Linden, A. S. L. Mal-abarba, A. J. Short, and A. Winter, “Equili-bration time scales of physically relevant observ-ables,” Phys. Rev. X7, 031027 (2017).

[4] M. Srednicki, “The approach to thermal equilib-rium in quantized chaotic systems,” J. Phys. A: Math. Gen.32, 1163 (1999).

[5] F. A. Pollock, C. Rodríguez-Rosario, T. Frauen-heim, M. Paternostro, and K. Modi, “Opera-tional markov condition for quantum processes,”

Phys. Rev. Lett.120, 040405 (2018).

[6] C. Arenz, R. Hillier, M. Fraas, and D. Bur-garth, “Distinguishing decoherence from alterna-tive quantum theories by dynamical decoupling,”

Phys. Rev. A92, 022102 (2015).

[7] M. A. Schlosshauer, Decoherence and the Quantum-To-Classical Transition (Springer Berlin Heidelberg, 2007).

[8] I. de Vega and D. Alonso, “Dynamics of non-Markovian open quantum systems,” Rev. Mod. Phys.89, 015001 (2017).

[9] F. Ciccarello, “Collision models in quantum op-tics,” Quantum Meas. Quantum Metrol. 4, 53 (2017).

[10] H. Spohn and J. L. Lebowitz, “Irreversible ther-modynamics for quantum systems weakly cou-pled to thermal reservoirs,” in Advances in Chemical Physics, edited by S. A. Rice (John Wiley & Sons, Ltd, 2007) pp. 109–142.

[11] M. Ledoux,The Concentration of Measure Phe-nomenon, Mathematical surveys and mono-graphs (American Mathematical Society, 2001).

[12] V. D. Milman and G. Schechtman, Asymp-totic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Mathematics No. 1200 (Springer-Verlag, 1986).

[13] S. Boucheron, G. Lugosi, and P. Massart, Con-centration Inequalities: A Nonasymptotic Theory of Independence (OUP Oxford, 2013).

[14] S. Popescu, A. J. Short, and A. Winter, “Entan-glement and the foundations of statistical me-chanics,” Nat. Phys.2, 754 (2006).

[15] Y. Li, “Seminar 6 of selected topics in mathematical physics: Quantum information theory,” http://www.mpmueller.net/seminar/ talk6.pdf(2013).

[16] L. Masanes, A. J. Roncaglia, and A. Acín, “Com-plexity of energy eigenstates as a mechanism for equilibration,” Phys. Rev. E87, 032137 (2013). [17] G. Chiribella, G. M. D’Ariano, and P. Perinotti,

“Quantum circuit architecture,”Phys. Rev. Lett. 101, 060401 (2008).

[18] G. Chiribella, G. M. D’Ariano, and P. Perinotti, “Theoretical framework for quantum networks,”

Phys. Rev. A80, 022339 (2009).

[19] F. Costa and S. Shrapnel, “Quantum causal mod-elling,” New J. Phys. 18, 063032 (2016).

[20] K. Modi, “Operational approach to open dynam-ics and quantifying initial correlations,” Sci. Rep. 2, 581 (2012).

[21] F. A. Pollock, C. Rodríguez-Rosario, T. Frauen-heim, M. Paternostro, and K. Modi, “Non-Markovian quantum processes: Complete frame-work and efficient characterization,” Phys. Rev. A97, 012127 (2018).

[22] H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vac-chini, “Colloquium: Non-Markovian dynamics in open quantum systems,” Rev. Mod. Phys. 88, 021002 (2016).

[23] M.-D. Choi, “Completely positive linear maps on complex matrices,” Linear Algebra Its Appl.10, 285 (1975).

[24] A. Jamiołkowski, “Linear transformations which preserve trace and positive semidefiniteness of operators,” Rep. Math. Phys.3, 275 (1972). [25] S. Milz, F. A. Pollock, and K. Modi, “An

in-troduction to operational quantum dynamics,”

Open Syst. Inf. Dyn. 24, 1740016 (2017). [26] A. J. Short and T. C. Farrelly, “Quantum

equili-bration in finite time,” New J. Phys.14, 013063 (2012).

[27] C. Gogolin, M. P. Müller, and J. Eisert, “Ab-sence of thermalization in nonintegrable sys-tems,” Phys. Rev. Lett. 106, 040401 (2011). [28] A. Gilchrist, N. K. Langford, and M. A. Nielsen,

“Distance measures to compare real and ideal quantum processes,” Phys. Rev. A 71, 062310 (2005).

groups: An application of Schur-Weyl duality,”

arXiv:1408.3782 (2014).

[30] Y. Gu,Moments of Random Matrices and. Wein-garten Functions, Master’s thesis, Queen’s Uni-versity, Ontario, Canada (2013).

[31] B. Collins and P. Śniady, “Integration with re-spect to the Haar measure on unitary, orthogonal and symplectic group,” Commun. Math. Phys. 264, 773 (2006).

[32] Z. Puchała and J. A. Miszczak, “Symbolic inte-gration with respect to the Haar measure on the unitary groups,” Bull. Pol. Acad. Sci. Tech. Sci. 65, 21 (2017).

[33] D. A. Roberts and B. Yoshida, “Chaos and com-plexity by design,” J. High Energy Phys. 2017, 121 (2017).

[34] F. Brandão, A. W. Harrow, and M. Horodecki, “Local random quantum circuits are approx-imate polynomial-designs,” Commun. Math. Phys.346, 397 (2016).

[35] T. Gläßle, “Seminar 8 of selected topics in mathematical physics: Quantum information theory,” http://www.mpmueller.net/seminar/ talk8.pdf(2013).

[36] E. Lubkin, “Entropy of an n-system from its cor-relation with a k-reservoir,” J. Math. Phys. 19, 1028 (1978).

[37] D. N. Page, “Average entropy of a subsystem,”

Phys. Rev. Lett.71, 1291 (1993).

[38] S. Lloyd and H. Pagels, “Complexity as thermo-dynamic depth,” Ann. Phys. New York188, 186 (1988).

[39] A. J. Scott and C. M. Caves, “Entangling power of the quantum baker’s map,” J. Phys. A: Math. Gen.36, 9553 (2003).

[40] O. Giraud, “Purity distribution for bipartite ran-dom pure states,” J. Phys. A: Math. Theor.40, F1053 (2007).

[41] A. D. Pasquale, P. Facchi, V. Giovannetti, G. Parisi, S. Pascazio, and A. Scardicchio, “Sta-tistical distribution of the local purity in a large quantum system,” J. Phys. A: Math. Theor.45, 015308 (2012).

[42] F. Mezzadri, “How to generate random matrices from the classical compact groups,” Notices of the AMS54, 592 (2007).

[43] J. Cotler, N. Hunter-Jones, J. Liu, and B. Yoshida, “Chaos, complexity, and random ma-trices,” J. High Energy Phys.2017, 48 (2017). [44] R. A. Low, “Large deviation bounds for

k-designs,” Proc. R. Soc. A465, 3289 (2009).

[45] Y. Nakata, C. Hirche, M. Koashi, and A. Win-ter, “Efficient quantum pseudorandomness with nearly time-independent Hamiltonian dynam-ics,” Phys. Rev. X7, 021006 (2017).

[46] S. Milz, F. Sakuldee, F. A. Pollock, and K. Modi, “Kolmogorov extension theorem for (quantum) causal modelling and general probabilistic theo-ries,” arXiv:1712.02589 (2017).

[47] F. Anza, C. Gogolin, and M. Huber, “Eigenstate thermalization for degenerate observables,”Phys. Rev. Lett.120, 150603 (2018).

[48] F. A. Pollock and K. Modi, “Tomographically re-constructed master equations for any open quan-tum dynamics,” Quantum2, 76 (2018).

[49] D. Tamascelli, A. Smirne, S. F. Huelga, and M. B. Plenio, “Nonperturbative treatment of non-Markovian dynamics of open quantum sys-tems,” Phys. Rev. Lett. 120, 030402 (2018). [50] I. A. Luchnikov, S. V. Vintskevich, H.

Ouer-dane, and S. N. Filippov, “Simulation complex-ity of open quantum dynamics: Connection with tensor networks,” Phys. Rev. Lett. 122, 160401 (2019).

[51] A. Arias, A. Gheondea, and S. Gudder, “Fixed points of quantum operations,” J. Math. Phys. 43, 5872 (2002).

[52] M. Cramer, “Thermalization under randomized local Hamiltonians,” New J. Phys. 14, 053051 (2012).

[53] D. Weingarten, “Asymptotic behavior of group integrals in the limit of infinite rank,” J. Math. Phys.19, 999 (1978).

[54] A. Ginory and J. Kim, “Weingarten calculus and the IntHaar package for integrals over compact matrix groups,” arXiv:1612.07641 (2016). [55] P. Taranto, K. Modi, and F. A. Pollock,

“Emergence of a fluctuation relation for heat in nonequilibrium Landauer processes,” Phys. Rev. E97, 052111 (2018).

[56] J. M. Epstein and K. B. Whaley, “Quantum speed limits for quantum-information-processing tasks,” Phys. Rev. A95, 042314 (2017).

[57] M. Deza and E. Deza,Encyclopedia of Distances, Encyclopedia of Distances (Springer Berlin Hei-delberg, 2009).

A

The process tensor

The Choi state of a process tensor with initial stateρis given by

Υk:0= trE[Uk:0(ρ⊗Ψ⊗k)Uk†:0], (14)

with

Uk:0≡(Uk⊗1)Sk· · ·(U1⊗1)S1(U0⊗1), (15)

where all identities are in the total ancillary system and the Ui are ES unitary operators at step i (here the

initial unitaryU0 is taken simply to randomize the initial state according to the Haar measure), and

Si≡X

α,β

Sαβ⊗1A1B1···Ai−1Bi−1 ⊗ |βihα| ⊗1BiAi+1Bi+1···AkBk, (16)

with Sαβ = 1E ⊗ |αihβ|. Notice that Sασ = S†σα, SabS†cd = δbdSac and tr(Sab) = dEδab. Also, Υ ∈

S(HSNk

i=1HAiBi), i.e. the resulting Choi process tensor state belongs to the S-ancillary system, which has dimensiond2_Sk+1.

B

Derivation of an upper-bound on non-Markovianity

As a first approach, given the complexity of computing (averages over) the Markovian Choi state, we may upper bound this distance by the one with respect to the maximally mixed state (the noisiest Markovian process possible)

N ≤D(Υ, d−(2_S k+1)₁). (17)

We may further bound this by considering the following cases separately.

1. Case dE < d2Sk+1: We notice that

5

rank(Υ) ≤ dE. Letting γ be the diagonal matrix of up to dE

non-vanishing eigenvaluesλγi of the Choi state, we may write

kΥ− 1

d2_Sk+1k1=

dE

X

i=1

λγi−

1

d2_Sk+1

+

d2k+1

S

X

j=dE+1

− 1

d2_Sk+1

=kγ− 1E

d2_Sk+1k1+ 1− dE

d2_Sk+1, (18)

where | · | denotes the standard absolute value, so using the inequality kXk1≤ p

dim(X)kXk2 for a square

matrixX (which can be derived from the Cauchy-Schwarz inequality applied to the eigenvalues of X), where

kXk2= p

tr(XX†_{)is the Schatten 2-norm,}

kΥ− 1

d2_Sk+1k1≤

p

dEkγ− 1E

d2_Sk+1k2+ 1− dE

d2_Sk+1

=

s

dEtr[Υ2] +

d2

E

d4_Sk+2 − 2dE

d2_Sk+1 + 1− dE

d2_Sk+1, (19)

Furthermore, applying Jensen’s inequality (in particular for the square-root,E[

√

X]≤p

E[X]) with expectation over evolution (either in the ergodic or time-independent case), this gives

E[N]≤ 1 2

s

dEE[tr(Υ2)] + d2

E

d4_Sk+2 − 2dE

d2_Sk+1 + 1− dE

d2_Sk+1

!

. (20)

2. Case dE ≥ d2Sk+1: This case is a small subsystem limit for most k. Directly applying kXk1≤

p

dim(X)kXk2as before,

kΥ− 1

d2_Sk+1k1≤

q

d2_Sk+1kΥ− 1

d2_Sk+1k2=

q

d2_Sk+1tr[Υ2_{]−1, (21)}

Similarly, taking the average over evolution, by means of Jensen’s inequality,

E[N]≤ 1 2

q

This proves that the function

Bk(dE, dS)≡



 

 

√

dEE[tr(Υ2_)]−x+y

2 if dE< d

2k+1

S

p

d2_Sk+1E[tr(Υ2_)]−1

2 if dE≥d

2k+1

S ,

(23)

with x ≡ dE

d2_Sk+1(1 +y), y ≡ 1− dE

d2_Sk+1, as given in the main text, provides an upper bound on the average

non-MarkovianityE[N].

C

Haar random averages

C.1

The Haar measure

The Haar measure is the unique probability measureµon any of the orthogonal, unitary or symplectic groups that is both left and right invariant, for our purposes meaning that R f(V U)dµ(U) = R f(U V)dµ(U) =

R

f(U)dµ(U) for all unitaries V, with normalization Rdµ(U) = 1. A random pure state can thus be gen-erated by taking|φi ∈ Hwithd= dim(H), sampling a unitary matrixU ∈U(d)according to the Haar measure and letting|φihφ| ≡φ7→U φU†. Its average over the Haar measure is then given by

EH(φ)≡

Z

U(d)

U φ U†dµ(U), (24)

which can also be seen as a so-called 1-fold twirl and be computed by the Schur-Weyl duality [29,33]. When we deal with averages over the Haar measure with several unitaries we will denote by _EUi and_EU the ergodic and time-independent cases, respectively.

C.2

Schur-Weyl duality and the moments of the unitary group

The Schur-Weyl duality allows to compute integrals oftwirled maps over the Haar measure given by

Φ(k)(X)7→

Z

U(d)

U⊗kX(U⊗k)†dµ(U), (25)

where X ∈ H⊗k_{; in particular, the caseU}†_{AU XU}†_{BU can be taken to a twirled form and be evaluated as in}

e.g. in [29], which we here use to compute the average purity of the Choi (ergodic) process tensor. In particular, Φ(k) commutes with all V⊗k such thatV ∈U(d), and by such property, the Schur-Weyl duality assures that also

Φ(k)(X) = X

σ∈Sk

fτ(X)Pτ, (26)

where the sum is over the symmetric group Sk on k symbols, Pτ is a permutation operator defined by

Pτ|x1, . . . , xki = |xτ(1), . . . , xτ(k)i and fτ is a linear function of X [33]. The k = 1 case is well known [15]

and in general for any operator acting on_Cd _{results in a completely depolarizing channel,}

Φ(1)(X) =E(X) = tr(X)

d 1. (27)

A related result, which can equivalently be used to compute integrals over the unitary group for polynomials in the unitaries with the Haar measure, and which we also make use of in this work, is the one for thek-moments of thed-dimensional unitary group [30],

Z

U(d)

k

Y

`=1

Ui`j`U

∗

i0 `j

0

`dµ(U) =

X

σ,τ∈Sk k

Y

`=1

δi`i0<sub>σ(`)</sub>δj`j<sub>τ</sub>0<sub>(`)</sub>Wg(τ σ

−1_{, d), (28)}

whereUij is the ij component ofU ∈U(d)and Wgis known as the Weingarten function (here specifically for

symmetric group; it is a fairly complicated function to evaluate explicitly and we refer to [29,30] for the details, in any case it is simply a rational function in the dimension argument d (particular cases are often given in the literature for smallk, see e.g. [30,33]). An alternative is to perform numerical calculations, as a computer package in the Mathematica software was developed in [32] that allows to numerically evaluate Eq.(28) and the Weingarten function (a subsequent version for Maple is also now available in [54]).

Here we obtain the expected states _E(Υ) and expected purities _E[tr(Υ2_{)] over the Haar measure of Choi}

process tensor statesΥas defined by Eq.(14). We compute these averages for both the ergodic_EUi and time-independent_EU cases. The time-independent averages are written in terms of theWgfunction; we express them explicitly for the simplest cases and we show that in the small subsystem limit they reduce to the corresponding ergodic averages.

The asymptotic behavior of Eq.(28)boils down to that of theWgfunction and in [30] it has been shown that

Wg(σ∈ Sn, d)∼ 1

d2n−#σ, asd→ ∞, (29)

as a refinement of a result in [31], where #σ is the number of cycles of the permutationσ counting also fixed points (assignments from an element to itself,σ(x) =x).

D

Average Processes

D.1

Ergodic average process Choi state

The ergodic average Choi state of akstep process is given by the maximally mixed state,

E_Ui(Υk:0) =

1

d2_Sk+11SA1B1···AkBk. (30)

This can be seen by a simple guess, as there arek+ 1 independent integrals of a 1-fold twirl kind, and both the initial state and theSoperators have trace one. This is also an intuitive result, here corresponding to the

noisiest process possible.

The detail of the calculation is as follows.

From the definition of the Choi process tensor state(14), in the ergodic case,

Υk:0= trE

h

(Uk⊗1)Sk· · ·(U1⊗1)S1(U0⊗1)(ρ⊗Ψ⊗k)(U0†⊗1)S † 1(U

†

1 ⊗1)· · · S †

k(U

†

k ⊗1)

i

= trE

X h

UkSαkβk· · ·U1Sα1β1U0ρU

†

0S

†

γ1δ1U

† 1· · ·S

†

γkδkU

†

k

i

⊗[(|β1ihα1| ⊗1B1)Ψ(|γ1ihδ1| ⊗1B1)⊗ · · · ⊗(|βkihαk| ⊗1Bk)Ψ(|γkihδk| ⊗1Bk)], (31)

where the sum is over repeated indices (Greek letters), hence by repeatedly evaluating the 1-fold twirl (27), withd=dEdS (also, the unitary groups are implicitly of dimensiondEdS), we get

EUi(Υk:0) = trE

X Z

Uk···U0

UkSαkβk· · ·U1Sα1β1U0ρU

†

0S

†

γ1δ1U

† 1· · ·S

†

γkδkU

†

k

⊗[(|β1ihα1| ⊗1B1)Ψ(|γ1ihδ1| ⊗1B1)⊗ · · · ⊗(|βkihαk| ⊗1Bk)Ψ(|γkihδk| ⊗1Bk)]

= 1

dk Sd

trE

X Z

Uk−1···U0

hekβk|Uk−1Sαk−1βk−1· · ·U1Sα1β1U0ρU

†

0S

†

γ1δ1U

† 1· · ·S

†

γk−1δk−1U

†

k−1|ekδki

1S⊗ |β1α1· · ·βkαkihδ1γ1· · ·δkαk|

= dE dk

Sd2

trE

X Z

Uk−2···U0

hek−1βk−1|Uk−2Sαk−2βk−· · ·2 U1Sα1β1U0ρU

†

0S

†

γ1δ1U

† 1· · ·S

†

γk−2δk−2U

†

k−2|ek−1δk−1i

1S⊗ |β1α1· · ·βk−1αk−1ihδ1γ1· · ·δk−1αk−1| ⊗1AkBk ..

.

= d

k+1

E

dk Sdk+1

1SA1B1···AkBk

= 1

d2_Sk+11SA1B1···AkBk, (32)

D.2

Time-independent average process Choi state

The average Choi process tensor forksteps in the time-independent case can be given in terms of a set{|s(0)_i i}dS si=1 ofS system bases fori= 0, . . . , k as

EU(Υk:0) =

1 dk S

X

σ,τ∈Sk+1

ρ_τ(0);0Wg(τ σ−1)∆(dE)

k,σ,τ|sσ(k)ihsk| k

O

j=1

|sσ(j−1)s0τ(j)ihsj−1s0j|, (33)

with implicit sum over all repeated basis (s(0)_i ) indices, where hereSk+1 is the symmetric group on{0, . . . , k},

and with the definitions

ρ_τ(0);0=he0_τ(0)s0_τ(0)|ρ|e0₀s0₀i, (34)

∆(dE)

k,σ,τ =δeσ(k)ek k

Y

`=1

δeσ(`−1)e0τ(`)δe`−1e0`, (35)

where{|e(0)_i i}dE

ei=1, withi= 0, . . . , k, also summed over all its elements, is a set of environment bases and the∆ term is simply a monomial indE with degree determined by σandτ.

The casek= 0recoversEU(Υ0:0) =1S/dS as expected, as no process occurs.

In the case of the superchannel k = 1, we get Eq. (41), which is quite different from the maximally mixed state that arises in the ergodic case, although it converges to it asdE→ ∞. We notice thattr[EU(Υ1:0)] = 1as expected and one may also verify the purity of this average state to be given by Eq.(42)which is in factclose

to that of the maximally mixed state for alldE,dS and1/dS ≤tr(ρ2S)≤1, and converges to it whendE → ∞

at aO(1/dE)rate.

The detail of the calculation is as follows. By definition,

EU(Υk:0) =

trE

dk S

XZ

U

USαkβk· · ·USα1β1U ρU

†_S†

γ1δ1U

†_{· · ·S}†

γkδkU

†_{dµ(U)⊗ |β}

1α1. . . βkαkihδ1γ1. . . δkγk|.

(36)

We now consider directly decomposing the unitaries as U =P_U

ab|aihb| andU† =P_U∗

a0_b0|b0iha0|–notice that theaandblabels refer to the wholeESspace–, introducing anEbasis in theSoperators asSab=P|eaiheb|,

and then evaluating thek+ 1 moments ofU(dEdS)by means of Eq.(28),

Z

U

USαkβk· · ·USα1β1U ρU

†_S†

γ1δ1U

†_{· · ·S}†

γkδkU

†_dµ(U)

=X

Z

U

Ui0j0· · ·UikjkU

∗

i0

0j00· · ·U

∗

i0 kj

0

kdµ(U)|ikihj0|ρ|j

0 0ihi0k|

k

Y

`=1

δj`(eα)`δ(eβ)`i`−1δj`0(e0γ)`δ(e0δ)`i0`−1

=X X

σ,τ∈Sk+1

hj0|ρ|j00iδi0i0_σ(0)δj0j_τ0₍₀₎ k

Y

`=1

δi`i0<sub>σ(`)</sub>δj`j<sub>τ</sub>0<sub>(`)</sub>δj`(eα)`δ(eβ)`i`−1δj<sub>`</sub>0(e0γ)`δ(e0δ)`i0`−1Wg(τ σ

−1_)|ikihi0

k|

=X X

σ,τ∈Sk+1

hj_τ0₍₀₎|ρ|j00iδiki0_σ(k) k

Y

`=1

δ(eβ)`i0<sub>σ(`</sub>−1)δ(e

0_δ)

`i0`−1δ(eα)`j<sub>τ</sub>0<sub>(`)</sub>δ(e0γ)`j`0Wg(τ σ

−1_)|ikihi0

k|, (37)

where hereSk+1 denotes the symmetric group on {0, . . . , k}, and which takingi→ς andj →0ς0 to recover

eachE andS part explicitly, turns into

Z

U

USαkβk· · ·USα1β1U ρU

†_S†

γ1δ1U

†_{· · ·S}†

γkδkU

†_dµ(U)

=X X

σ,τ∈Sk+1

h0_τ(0)ς_τ0₍₀₎|ρ|0₀ς₀0i k

Y

`=1

δσ(`−1)τ0(`)δ`−1

0

`δβ`ςσ(`−1)δδ`ς`−1δα`ςτ0(`)δγ`ς

0 `Wg(τ σ

−1_)|

σ(k)ςσ(k)ihkςk|,

and thus

EU(Υk:0) =

P

dk S

X

σ,τ∈Sk+1

h0_τ(0)ς_τ0₍₀₎|ρ|00ς00iWg(τ σ−1)δσ(k)k k

Y

`=1

δσ(`−1)0<sub>τ(`)</sub>δ`−10`|ςσ(k)ςσ(0)ς

0

τ(1)· · ·ςσ(k−1)ςτ0(k)ihςkς0ς10· · ·ςk−1ςk0|,

(39)

as stated by Eq.(33).

D.2.1 Superchannel case

For the superchannel case,k= 1, we haveS2={(0,1),(0)(1)}where the elements are permutations stated in

cycle notation, representing the assignments (0,1) =

(

0→1

1→0 and (0)(1) =1

2 ₌

(

0→0

1→1 , then (we write α, β, γ, δ forς0, ς1, ς00, ς10, respectively to ease the notation)

EU(Υ1:0) =

P

dS

hδ|ρS|γi|αβγihβαδ|+d2E|βαδihβαδ|

Wg[12] +dE[hδ|ρS|γi|βαγihβαδ|+|αβδihβαδ|] Wg[(0,1)]

,

(40)

so that withWg[12, d] =_d21₋₁ and Wg[(0,1), d] =

−1

d(d2₋₁₎ [30] we get

EU(Υ1:0) =

1 d2

Ed

2

S−1

_d2

E

dS1

SAB+swap

dS

⊗ρ|_S −swap

dS

⊗1B

dS

−1SA

d2

S ⊗ρ|_S

, (41)

where _swap=P

i,j|ijihji| is the usual swap operator, and hence for the corresponding purity one may verify

that

tr[_EU(Υ1:0)2] =

2 (d2

Ed

2

S−1)2

₁

d3

S

+ tr(ρ2_S)d

2

S−dS−1

2d2

S

−d

2

E

dS

+d

4

EdS

2

. (42)

D.2.2 Small subsystem limit

When looking at the limitdE→ ∞, the term that does not vanish is the one withσ, τ =1k+1, i.e. with both

per-mutations being identities, as these generate the most numerator powers indEvia theδterms (correspondingly,

∆(dE)

k,σ,τ in Eq.(35)) when summed over allei’s, i.e.

dE

X

e(_i0)=1

e0_6=e0

0

δkk k

Y

`=1

δ`−10`δ`−10` =d k+1

E , (43)

all other permutations will vanish because of thedE powers in the denominator generated by theWgfunction,

the least powers produced by it are those whenστ−1=1k+1 _because#1n_{= #[(1)(2)· · ·(n)] =n, i.e. identity}

produces the greatest number of cycles, being the number of fixed points. FinallyP

0

0ς00h

0

0ς00|ρ|00ς00i= trρ= 1,

and thus

EU(Υk:0)∼ dk_E+1

dk S

Wg(1k+1)₁SA1B1...AkBk=

1

d2_Sk+11SA1B1...AkBk, whendE→ ∞, (44)

matching the average over an ergodic process.

E

Proof of Main Result

E.1

Preamble: Concentration of Measure (Levy’s lemma)

A functionf :X →Y between metric spaces(X, δ_X)and(Y, δ_Y)isL-Lipschitz if there is a real constantη≥0, known as the Lipschitz constant, such that